Envy-Free Allocation of Indivisible Goods via Noisy Queries

2026-02-06

AI Summary

This work addresses the problem of achieving envy-free allocations of indivisible goods when the valuations of two agents can only be accessed through noisy queries corrupted by Gaussian noise. The authors propose a non-adaptive query strategy coupled with a threshold-based allocation algorithm and, by combining probabilistic analysis with information-theoretic lower bounds, establish the first tight bounds on the query complexity for approximate envy-freeness in this noisy setting. Their results show that when the allowed envy level Δ satisfies Δ ≫ m^{1/4}, the optimal query complexity is Θ(m^{2.5}/Δ²) (up to logarithmic factors). The upper bound is achievable via a polynomial-time algorithm, while the lower bound holds for any adaptive querying strategy.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We introduce a problem of fairly allocating indivisible goods (items) in which the agents'valuations cannot be observed directly, but instead can only be accessed via noisy queries. In the two-agent setting with Gaussian noise and bounded valuations, we derive upper and lower bounds on the required number of queries for finding an envy-free allocation in terms of the number of items, $m$, and the negative-envy of the optimal allocation, $\Delta$. In particular, when $\Delta$ is not too small (namely, $\Delta \gg m^{1/4}$), we establish that the optimal number of queries scales as $\frac{\sqrt m }{(\Delta / m)^2} = \frac{m^{2.5}}{\Delta^2}$ up to logarithmic factors. Our upper bound is based on non-adaptive queries and a simple thresholding-based allocation algorithm that runs in polynomial time, while our lower bound holds even under adaptive queries and arbitrary computation time.

Problem

• envy-free allocation
• indivisible goods
• noisy queries
• fair division
• valuation uncertainty

Innovation

• envy-free allocation
• noisy queries
• indivisible goods
• query complexity
• fair division